Liquid and crystalline phases can be formed in so-called complex plasmas—plasmas enriched with solid particles in the nano- to micrometre range. The particles absorb electrons and ions and charge negatively up to a few volts. Due to their high mass compared to that of electrons and ions the particles dominate the processes in the plasma and can be observed on the most fundamental—the kinetic level. Through the strong Coulomb interaction between the particles it is possible that the particle clouds form fluid and crystalline structures. The latter is called `plasma crystal'. In this review we present an overview on the physics of this new area in plasma physics on the basis of theory and dedicated experiments performed in the laboratory and under microgravity conditions on the International Space Station.